I hate to be writing the very first comment in this thread without any actual help within, but it has been resting for a while now...

I noticed that

May be rewritten as (quite unusual form)

I don't have a clue whether there can be a slightest use of this though, sorry...

January 15th 2011, 03:59 PM

orange gold

Re-wrote it as:

I don't know where I'm going with this, but maybe that helped :D

January 15th 2011, 06:31 PM

tonio

Quote:

Originally Posted by orange gold

Re-wrote it as:

Your left side here doesn't equal the OP's left side: it must be a quartic pol. in x, and you wrote a quintic.

Tonio

I don't know where I'm going with this, but maybe that helped :D

.

January 16th 2011, 12:32 AM

Opalg

Quote:

Originally Posted by alexmahone

Find all integral solutions of the equation .

My attempt: The equation can be rewritten as .

Can someone please tell me how to proceed?

Let . Check that

Then , so it follows from (1) that is always too small to be a square root for

Next, , and the only integers for which this is not positive are and . For all other integers, is positive, and it follows from (3) that is too large to be a square root of

Thus, unless or , the only possible candidate for an integral square root of is , and it follows from (2) that this solution will only work if

Therefore the only values of that might give solutions to the problem are and . If then , which is not a square. But the other three values of give the solutions to the problem, namely .

January 16th 2011, 03:26 AM

roninpro

Quote:

Originally Posted by Opalg

Let . Check that

Then , so it follows from (1) that is always too small to be a square root for

Next, , and the only integers for which this is not positive are and . For all other integers, is positive, and it follows from (3) that is too large to be a square root of

Thus, unless or , the only possible candidate for an integral square root of is , and it follows from (2) that this solution will only work if

Therefore the only values of that might give solutions to the problem are and . If then , which is not a square. But the other three values of give the solutions to the problem, namely .

Could you elaborate a little bit more on why this method is exhaustive?

January 16th 2011, 12:41 PM

Opalg

Quote:

Originally Posted by roninpro

Quote:

Originally Posted by Opalg

Let . Check that

Then , so it follows from (1) that is always too small to be a square root for

Next, , and the only integers for which this is not positive are and . For all other integers, is positive, and it follows from (3) that is too large to be a square root of

Thus, unless or , the only possible candidate for an integral square root of is , and it follows from (2) that this solution will only work if

Therefore the only values of that might give solutions to the problem are and . If then , which is not a square. But the other three values of give the solutions to the problem, namely .

Could you elaborate a little bit more on why this method is exhaustive?

The strategy is quite simple really. We want to find an integer that is a square root for . My claim is that the best candidate for this square root is . Equation (1) shows that is definitely too small to be a square root for Similarly, equation (3) shows that is too large to be a square root for except possibly when or So, for all other values of , the square root must be bigger than and smaller than , which leaves as the only remaining possibility. But equation (2) shows that for to be the square root, it is necessary that , and that only happens when

January 16th 2011, 01:06 PM

chiph588@

Quote:

Originally Posted by Opalg

The strategy is quite simple really. We want to find an integer that is a square root for . My claim is that the best candidate for this square root is . Equation (1) shows that is definitely too small to be a square root for Similarly, equation (3) shows that is too large to be a square root for except possibly when or So, for all other values of , the square root must be bigger than and smaller than , which leaves as the only remaining possibility. But equation (2) shows that for to be the square root, it is necessary that , and that only happens when

Just to pick your brain a bit, what led you to choose in the first place?